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A332612 a(n) = Sum_{ i=2..n-1, j=1..i-1, gcd(i,j)=1 } (n-i)*(n-j).

%I A332612 #29 Jun 26 2023 02:04:30
%S A332612 0,0,2,11,32,77,148,268,442,691,1018,1472,2036,2780,3686,4786,6100,
%T A332612 7724,9598,11863,14454,17437,20818,24772,29172,34200,39794,46071,
%U A332612 52986,60817,69314,78860,89292,100720,113122,126686,141244,157294,174566,193228,213172,234954,258058,283189,309946,338473,368782,401516,436040
%N A332612 a(n) = Sum_{ i=2..n-1, j=1..i-1, gcd(i,j)=1 } (n-i)*(n-j).
%C A332612 Related to the number of linear dichotomies on a square grid.
%C A332612 A331771(n) = 8*a(n) + 4*n*(n-1) + 4*(n-1)^2.
%H A332612 Seiichi Manyama, <a href="/A332612/b332612.txt">Table of n, a(n) for n = 1..1000</a>
%H A332612 Jack Koplowitz, Michael Lindenbaum, and A. Bruckstein, <a href="https://www.researchgate.net/publication/3077520_Number_of_digital_straight_lines_on_an_N_N_grid">The number of digital straight lines on an NxN grid</a>, IEEE Transactions on Information Theory 36.1 (1990): 192-197. (See I_1.)
%F A332612 a(n) = (Sum_{i=2..n-1} (n-i)*(2n-i)*phi(i))/2. - _Chai Wah Wu_, Aug 17 2021
%p A332612 I1 := proc(n) local a, i, j; a:=0;
%p A332612 for i from 2 to n-1 do for j from 1 to i-1 do
%p A332612 if igcd(i,j)=1 then a := a+(n-i)*(n-j); fi; od; od; a; end;
%p A332612 [seq(I1(n),n=1..40)];
%o A332612 (PARI) a(n) = sum(i=2, n-1, sum(j=1, i-1, if (gcd(i,j)==1, (n-i)*(n-j)))); \\ _Michel Marcus_, Mar 14 2020
%o A332612 (Python)
%o A332612 from sympy import totient
%o A332612 def A332612(n): return sum(totient(i)*(n-i)*(2*n-i) for i in range(2,n))//2 # _Chai Wah Wu_, Aug 17 2021
%Y A332612 The following eight sequences are all essentially the same. The simplest is A115004(n), which we denote by z(n). Then A088658(n) = 4*z(n-1); A114043(n) = 2*z(n-1)+2*n^2-2*n+1; A114146(n) = 2*A114043(n); A115005(n) = z(n-1)+n*(n-1); A141255(n) = 2*z(n-1)+2*n*(n-1); A290131(n) = z(n-1)+(n-1)^2; A306302(n) = z(n)+n^2+2*n. The present sequence and A331771 could be added to this list.
%K A332612 nonn
%O A332612 1,3
%A A332612 _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 14 2020